Existence Results for Generalized Vector Quasi-Equilibrium Problems in Hadamard Manifoldsâ€
Round 1
Reviewer 1 Report
Please revise the manuscript as per the attached report.
Comments for author File: 
Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 2 Report
In this manuscript several versions of the existence theorems are established for generalized vector quasi-equilibrium problems, in locally compact and σ-compact spaces without the need to use continuity conditions. The main result is the fixed point theorem on a product manifold of Hadamard, Theorem 3.3. Later on, the author applies the results obtained to saddle point and minimax problems.
The work is well written and organized. I believe that the results obtained are correct and novel, and the symbology is appropriate.
Although the applications presented show the scope and strength of the results obtained, the absence of a Conclusions section, where the advantages and scope over the results known from the literature are shown, does not allow an adequate evaluation of the general impact of the work.
Author Response
Dear Reviewer:
Thank you very much for your valuable comments on the manuscript.
The responses to the specific comments are as follows:
Although the applications presented show the scope and strength of the results obtained, the absence of a Conclusions section, where the advantages and scope over the results known from the literature are shown, does not allow an adequate evaluation of the general impact of the work.
Response:
The Conclusions section has been added in the revised manuscript (Section 6) and is highlighted in blue color for your convenience of re-reviewing.
In addition, a paragraph on the importance of equilibrium problems on the manifolds has been added to the Introduction.
Author Response File: Author Response.pdf
Reviewer 3 Report
In this paper, author established manifold versions of existence theorems
for generalized vector quasi-equilibrium problems in locally compact and
$\sigma$-compact spaces without any continuity assumption. The results presented in the paper seem correct, and the techniques employed tosolve the problems are generally standard with some novelty. However, the following points should be considered to improve thepresentation of this paper:
1- In the introduction part, the author should give more background works in details about Hadamard manifolds
2- Grammatical error and some typos exist that should be checked and corrected throughout the paper.
3-Mentioned the main contributions in the end of introduction part.
4-what are the advantages of Hadamard manifolds
5-Add example to illustrate your results
6- Conclusion part is missed.
7-Few remarks can be added to strengthen the ideas of this paper. What are the additional way in which the manuscript could be improved?
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 4 Report
The author consider : the fixed point problem , equilibrium problem and minimax problem for convex valued mappings of subsets in some top[ological vector spaces. There are proved 4 Theorems about the above problems. The obtained results are new but not so much interesting . The proofs are correct . The paper is well organized.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 5 Report
see the report
Comments for author File: Comments.pdf
Author Response
Dear Reviewer,
Thank you very much for your valuable comments on the manuscript. The revised parts are highlighted in blue color for your convenience of re-reviewing. Thank you very much again.
The responses to the specific comments are as follows:
- In whole paper put for all instead ∀. Hence without any logical symbol.
Response: All symbols ∀ have been replaced by "for all" on pp. 1--3, 6, 10, 12, 13, 14.
- Further, add the CONCLUSION at end of the paper as in each MDPI type paper.
Response: The Conclusions section 6 has been added in the revised manuscript.
- Lacks of recent items
Response: The references [9] and [20] have been added.
In addition, a paragraph on the importance of equilibrium problems on the manifolds has been added to the Introduction.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
The revised manuscript is suitable for publication.
